Incorporating Control Motion

Let
denote the vertical position of the mass
in Fig.9.22. (We still assume
.) We can think of
as the position of the control point on the
plectrum, e.g., the position of the ``pinch-point'' holding the
plectrum while plucking the string. In a harpsichord,
can be
considered the jack position [350].

Also denote by
the rest length of the spring
in Fig.9.22, and let
denote the
position of the ``end'' of the spring while not in contact with the
string. Then the plectrum makes contact with the string when

where
denotes string vertical position at the plucking point
. This may be called the collision detection equation.

Let the subscripts
and
each denote one side of the scattering
system, as indicated in Fig.9.23. Then, for example,
is the displacement of the string on the left (side
) of plucking point, and
is on the right side of
(but
still located at point
). By continuity of the string, we have

When the spring engages the string (
) and begins to compress,
the upward force on the string at the contact point is given by

where again
. The force
is applied given
(spring is in contact with string) and given
(the force
at which the pluck releases in a simple max-force
model).10.15 For
or
the applied force is zero and the entire
plucking system disappears to leave
and
,
or equivalently, the force reflectance becomes
and the
transmittance becomes
.

During contact, force equilibrium at the plucking point requires
(cf. §9.3.1)

This system is diagrammed in Fig.9.24. The manipulation of the
minus signs relative to Fig.9.23 makes it convenient for
restricting
to positive values only (as shown in the
figure), corresponding to the plectrum engaging the string going up.
This uses the approximation
,
which is exact when
, i.e., when the plectrum does not affect
the string displacement at the current time. It is therefore exact at
the time of collision and also applicable just after release.
Similarly,
can be used to trigger a release of the
string from the plectrum.